Chapter 8

Maximize \(z=x_{1}+2 x_{2}+3 x_{3}$$$x_{1}+x_{2}+x_{3} \leq 12$$ subject to \)2 x_{1}+x_{2}+3 x_{3} \leq 18$$x_{1}, x_{2}, x_{3} \geq 0$

\(\begin{aligned} \text { Maximize } z=& x_{1}+2 x_{2}+x_{3} \\ & x_{1}+x_{2} \leq 3 \\ \text { subject to } & x_{2}+x_{3} \leq 4 \\ & x_{1}+x_{3} \leq 5 \end{aligned}\)

A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and \(\$ 20\) of capital, and each acre of corn requires 16 hours of labor and \(\$ 40\) of capital. The farmer has at most 800 hours of labor and \(\$ 2400\) of capital available. If the profit from an acre of wheat is \(\$ 80\) and from an acre of corn is \(\$ 100,\) how many acres of each crop should she plant to maximize her profit?

A factory manufactures chairs, tables and bookcases each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 600 hours; the second at most 500 hours; and the third at most 300 hours. A chair requires 1 hour of cutting, 1 hour of assembly, and 1 hour of finishing; a table needs 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; and a bookcase requires 3 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is \(\$ 20\) per unit for a chair, \(\$ 30\) for a table, and \(\$ 25\) for a bookcase, how many units of each should be manufactured to maximize profit?

The Acme Apple company sells its Pippin, Macintosh, and Fuji apples in mixes. Box I contains 4 apples of each kind; Box II contains 6 Pippin, 3 Macintosh, and 3 Fuji; and Box III contains no Pippin, 8 Macintosh and 4 Fuji apples. At the end of the season, the company has altogether 2800 Pippin, 2200 Macintosh, and 2300 Fuji apples left. Determine the maximum number of boxes that the company can make.

In problems 1-2, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method. $$ \begin{aligned} &\begin{aligned} \text { Minimize } z=& 6 x_{1}+8 x_{2} \\ & 2 x_{1}+3 x_{2} \geq 7 \end{aligned}\\\ &\text { subject to }\\\ &4 x_{1}+5 x_{2} \geq 9\\\ &x_{1}, x_{2} \geq 0 \end{aligned} $$

convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method. $$ \begin{array}{l} \text { Minimize } z=5 x_{1}+6 x_{2}+7 x_{3} \\ \text { subject to } \begin{aligned} & 3 x_{1}+2 x_{2}+3 x_{3} \geq 10 \\ & 4 x_{1}+3 x_{2}+5 x_{3} \geq 12 \\ x_{1}, x_{2}, x_{3} \geq & 0 \end{aligned} \end{array} $$

In the next two problems, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method. $$ \begin{aligned} &\begin{aligned} \text { Minimize } z=& 4 x_{1}+3 x_{2} \\ & x_{1}+x_{2} \geq 10 \end{aligned}\\\ &\text { subject to }\\\ &3 x_{1}+2 x_{2} \geq 24\\\ &x, x_{2} \geq 0 \end{aligned} $$

A diet is to contain at least 8 units of vitamins, 9 units of minerals, and 10 calories. Three foods, Food A, Food B, and Food C are to be purchased. Each unit of Food A provides 1 unit of vitamins, 1 unit of minerals, and 2 calories. Each unit of Food \(\mathrm{B}\) provides 2 units of vitamins, 1 unit of minerals, and 1 calorie. Each unit of Food C provides 2 units of vitamins, 1 unit of minerals, and 2 calories. If Food A costs \(\$ 3\) per unit, Food B costs \(\$ 2\) per unit and Food C costs \(\$ 3\) per unit, how many units of each food should be purchased to keep costs at a minimum?

Solve the following linear programming problems using the simplex method. $$ \begin{aligned} &\begin{array}{l} \text { Maximize } z=5 x_{1}+3 x_{2} \\ \qquad x_{1}+x_{2} \leq 12 \end{array}\\\ &\text { subject to }\\\ &2 x_{1}+x_{2} \leq 16\\\ &x_{1} \geq 0 ; x_{2} \geq 0 \end{aligned} $$

Solve the following linear programming problems using the simplex method. Maximize \(z=5 x_{1}+8 x_{2}\) subject to \(x_{1}+2 x_{2} \leq 30\) \(3 x_{1}+x_{2} \leq 30\) \(x_{1} \geq 0 ; x_{2} \geq 0\)

Solve the following linear programming problems using the simplex method. Maximize \(z=2 x_{1}+3 x_{2}+x_{3}\) $$ 4 x_{1}+2 x_{2}+5 x_{3} \leq 32 $$ subject to \(2 x_{1}+4 x_{2}+3 x_{3} \leq 28\) \(x_{1}, x_{2}, x_{3} \geq 0\)

Solve the following linear programming problems using the simplex method. \(\begin{aligned} \text { Maximize } z=& x_{1}+6 x_{2}+8 x_{3} \\ & x_{1}+2 x_{2} \leq 1200 \\ \text { subject to } 2 x_{2}+x_{3} & \leq 1800 \\ & 4 x_{1}+x_{3} \leq 3600 \\ x_{1}, x_{2}, x_{3} \geq 0 & \end{aligned}\)

Solve the following linear programming problems using the simplex method. \(\begin{aligned} \text { Maximize } z=& 6 x_{1}+8 x_{2}+5 x_{3} \\ & 4 x_{1}+x_{2}+x_{3} \leq 1800 \\ \text { subject to } & 2 x_{1}+2 x_{2}+x_{3} \leq 2000 \\ & 4 x_{1}+2 x_{2}+x_{3} \leq 3200 \\ x_{1}, x_{2}, x_{3} \geq & 0 \end{aligned}\)

Solve the following linear programming problems using the simplex method. Minimize \(z=12 x_{1}+10 x_{2}\) \(x_{1}+x_{2} \geq 6\) subject to \(2 x_{1}+x_{2} \geq 8\) \(x_{1} \geq 0 ; x_{2} \geq 0\)

Solve the following linear programming problems using the simplex method. Minimize \(z=4 x_{1}+6 x_{2}+7 x_{3}\) $$ x_{1}+x_{2}+2 x_{3} \geq 20 $$ subject to $$ \begin{array}{r} x_{1}+2 x_{2}+x_{3} \geq 30 \\ x_{1}, x_{2}, x_{3} \geq 0 \end{array} $$

Solve the following linear programming problems using the simplex method. Minimize \(z=40 x_{1}+48 x_{2}+30 x_{3}\) subject to \(\begin{aligned} & 2 x_{1}+2 x_{2}+x_{3} \geq 25 \\ & x_{1}+3 x_{2}+2 x_{3} \geq 30 \\ x_{1}, x_{2}, x_{3} \geq & 0 \end{aligned}\)

A department store sells three different types of televisions: small, medium, and large. The store can sell up to 200 sets a month. The small, medium, and large televisions require, respectively, 3,6 , and 6 cubic feet of storage space, and a maximum of 1,020 cubic feet of storage space is available. The three types, small, medium, and large, take up, respectively, \(2,2,\) and 4 sales hours of labor, and a maximum of 600 hours of labor is available. If the profit made from each of these types is \(\$ 40, \$ 80,\) and \(\$ 100,\) respectively, how many of each type of television should be sold to maximize profit, and what is the maximum profit?

A factory manufactures three products, \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C} .\) Each product requires the use of two machines, Machine I and Machine II. The total hours available, respectively, on Machine I and Machine II per month are 180 and 300 . The time requirements and profit per unit for each product are listed below. $$ \begin{array}{|l|l|l|l|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} \\ \hline \text { Machine I } & 1 & 2 & 2 \\ \hline \text { Machine II } & 2 & 2 & 4 \\ \hline \text { Profit } & 20 & 30 & 40 \\ \hline \end{array} $$

A company produces three products, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C},\) at its two factories, Factory I and Factory II. Daily production of each factory for each product is listed below. $$ \begin{array}{|l|l|l|} \hline & \text { Factory I } & \text { Factory II } \\ \hline \text { Product A } & 10 & 20 \\ \hline \text { Product B } & 20 & 20 \\ \hline \text { Product C } & 20 & 10 \\ \hline \end{array} $$ The company must produce at least 1000 units of product A, 1600 units of \(\mathrm{B},\) and 700 units of C. If the cost of operating Factory I is \(\$ 4,000\) per day and the cost of operating Factory II is \(\$ 5000\), how many days should each factory operate to complete the order at a minimum cost, and what is the minimum cost?

For his classes, Professor Wright gives three types of quizzes, objective, recall, and recall-plus. To keep his students on their toes, he has decided to give at least 20 quizzes next quarter. The three types, objective, recall, and recall-plus quizzes, require the students to spend, respectively, 10 minutes, 30 minutes, and 60 minutes for preparation, and Professor Wright would like them to spend at least 12 hours( 720 minutes) preparing for these quizzes above and beyond the normal study time. An average score on an objective quiz is \(5,\) on a recall type \(6,\) and on a recall-plus 7 , and Dr. Wright would like the students to score at least 130 points on all quizzes. It takes the professor one minute to grade an objective quiz, 2 minutes to grade a recall type quiz, and 3 minutes to grade a recall-plus quiz. How many of each type should he give in order to minimize his grading time?